An application of coding theory to a problem

8/7/2019 An application of coding theory to a problem

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Arch. M ath., Vol. 65, 46 1-4 64 (1995) 0003-889X/95/6505-0461 $ 2.30/09 1995 Birkhfiuser Verlag, Basel

An appl ica t ion of cod ing theory to a p rob lemin g raph i ca l enum era t i onBy

DIETER JUNGNICKEL an d SCOTT A. VANSTONE*)

I n t h i s n o t e w e e x p l o i t a r e l a t i o n s h i p b e t w e e n g r a p h t h e o r y a n d c o d i n g t h e o r y t oo b t a i n a v e r y s h o r t a n d e l e g a n t p r o o f o f R e a d ' s t h e o r e m g i v in g th e g e n e r a t i n g f u n c t i o nf o r t h e n u m b e r o f E u l e r i a n g r a p h s w i t h p v e r t i c e s a n d a n a n a l o g o u s ( t o o u r k n o w l e d g en e w ) t h e o r e m c o n c e r n i n g b i p a r t i t e E u l e r i a n g r a p h s . L e t u s r e c a l l t h e n e c e s s a r y b a c k -g r o u n d .

L e t G = (V , E ) b e a f in i te g r a p h w i th v e r t e x s e t V a n d e d g e s e t E . W e le t p = I V ] a n dq = [E l . An even subgrap h o f G is a s p a n n i n g s u b g r a p h o f G i n w h i c h e a c h v e r t e x h a s e v e nd e g re e . I t is w e ll k n o w n t h a t t h e s e t o f a ll e v e n s u b g r a p h o f G f o r m s a v e c t o r s p a c e u n d e rt h e s y m m e t r i c d if f er e n ce o f s u b g r a p h s ( w h e r e s u b g r a p h s a r e s i m p l y c o n s i d e r e d a s s u b s e t so f E ) . W e w i l l d e n o t e t h i s v e c to r s p a c e b y C( G ) a n d c o n s id e r i t a s a b in a r y l i n e a r c o d e .( T h e r e a d e r i s r e f e r r e d t o M a c W i l l i a m s a n d S lo a n e ( 1 9 7 7 ) a n d v a n L in t ( 1 9 8 2 ) f o rb a c k g r o u n d f r o m c o d i n g th e o ry . ) N o t e t h a t C ( G ) i s a s u b s p a c e o f t h e v e c t o r s p a c e f o r m e db y a l l s p a n n i n g s u b g r a p h s o f G w h i c h is e a si ly s ee n t o b e i s o m o r p h i c t o t h e q - d i m e n s i o n a lv e c t o r s p a c e V ( q , 2 ) o f q - t u p l e s w i th e n t r i e s f r o m G F ( 2 ) . I n t h i s i n t e r p r e t a t i o n , w ec o n s id e r t h e c o o r d in a t e p o s i t i o n s t o b e i n d e x e d b y t h e e d g e s o f G (i n s o m e fi x e d o r d e r in g ) ;t h e n e a c h s u b g r a p h i s a s s o c i a t e d w i t h t h e c o r r e s p o n d i n g ( b i na r y ) c h a r a c t e r is t i c v e c t o r o fl e n g th q (w h ic h h a s a n e n t r y I i n p o s i t i o n e i f a n d o n ly i f e b e lo n g s t o t h e g iv e n s u b g r a p h ) .B y a b u s e o f n o t a t i o n , w e w i l l d e n o t e t h e s u b s p a c e o f V ( q , 2 ) a s s o c i a t e d w i th a l l e v e ns u b g r a p h s o f G a g a in b y C ( G ) .

T h e v e c t o r s p a c e C ( G ) is ( in e i t h e r i n t e r p r e t a t i o n ) u s u a l l y c a l l e d t h e cycle space o f G ;i t s d im e n s io n i s k n o w n to b e q - p + 1 p r o v id e d t h a t G is c o n n e c t e d . I t i s c l e a r t h a t t h em in im u m w e ig h t o f a v e c to r i n C( G ) i s t h e s m a l l e s t c a r d in a l i t y o f a c y c l e i n G , i . e . t h eg i r t h 9 o f G . W e th u s h a v e t h e f o l l o w in g w e l l k n o w n r e s u l t .

P r o p o s i t i o n 1 . Let G be a connected graph with q edges on p vert ices , and let 9 be theg ir th o f G. Then C ( G ) is a bina ry [q, q - p + I, 9]-code.

*) This note was written w hile the first autho r was visiting the Dep artm ent of Combinatorics andOptimization of the University of Waterloo as an Adjunct Professor. He would like to thank hiscolleagues there for their hospitality. T he second autho r acknowledges the suppo rt of the N ation alScience and Engineering Research Counci! of Canada given under grant ~ 0GP0009258.


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46 2 D. JUNGNICKELan d S. A . VANSTONE ARCH. MATH.We sha l l he r eaf te r re fe r to C (G) as the (even) graphical code o f G . T h e s e c o d e s h a v e b e e n

s y s t e m a t i c a l l y s t u d i e d b y B r e d e s o n a n d H a k i m i ( 19 67 ), H a k im i a n d B r e d e s o n ( 19 68 ) a n dJ u n g n ic k e l a n d Va n s to n e ( 19 95 ); i n t h e s e p a p e r s , t h e p r o b l e m s o f e x t e n d in g e v e n g r a p h -i c a l c o d e s t o l a r g e r c o d e s a n d o f d e v i s in g ef f ic i en t d e c o d in g p r o c e d u r e s h a v e b e e n c o n s id -e r ed . W e r e m a r k t h a t t h e d u a l o f C ( G ) (i.e ., t h e o r t h o g o n a l c o m p l e m e n t i n V(q, 2) withr e s p e c t t o t h e s t a n d a r d i n n e r p r o d u c t ) i s, o f c o u r s e , n o th in g b u t t h e b o n d s p a c e o f G , s e ee .g . B o n d y a n d M u r ty ( 1 9 7 6 ) .

W e s h a l l r e q u i r e o n e s t a n d a r d i t e m f r o m c o d i n g t h e o r y . G i v e n a n y b i n a r y c o d e C o fl e n g t h q , o n e d e n t i n e s t h e ( h o m o g e o u s ) w e i g h t e n u m e r a t o r o f C a s t h e p o l y n o m i a l

qA c (x , y ) = ~ , A i x q - i y i,n=0

wh e r e A i d e n o te s t h e n u m b e r o f v e c to r s w i th we ig h t i ( i. e., w i th e x a c t l y i e n t r i e s 1 )in C . I f C i s a l i n e a r c o d e w i th d im e n s io n k , t h e n t h e w e l l - k n o w n M a c W i l l i a ms id e n t i ti e sa s s e r t t h a t t h e w e i g h t e n u m e r a t o r o f t h e d u a l c o d e C o f C i s g i v e n b y

1Ac l ( X ,y ) = ~ A c ( x + y , x - y ) ,s e e M a c W i l l i a ms a n d S lo a n e ( 1 9 7 7 ) .

T h e s e f e w r e m a r k s a l r e a d y s u f f ic e t o g iv e a s imp le a n d e l e g a n t c o d in g t h e o r e t i c d e r iv a -t i o n f o r t h e g e n e r a t i n g f u n c t io n f o r t h e n u m b e r o f l a b e l e d e u l e r i a n g r a p h s w i th p ve r t ic e s ,a r e s u l t d u e t o R e a d ( 1 9 6 2 ) ; s e e a l s o Ha r a r y a n d P a lme r ( 1 9 7 3 ) .

T h e o r e m 2. The polynomial wp (x) which has as the coefficient of x q the number oflabeled Eulerian graphs with p vertices and q edges, is given by

1 x)(~) Z \ n J \ 1 + x /wv(x) = p ( l + .= oP r o o f . N o te t h a t t h e E u le r i a n g r a p h s o n p v e r t i c e s a r e p r e c i s e ly t h e e v e n s u b g r a p h s

o f t h e c o m p l e t e g r a p h K p . T h u s Wp(X) i s n o t h i n g b u t t h e ( n o n - h o m o g e n e o u s ) w e i g h te n u m e r a t o r o f t h e e v e n g r a p h i c a l c o d e C - - C ( K p ). U s i n g t h e M a c W i U i a m s i d e nt it ie s , w em a y i n s t e a d c o m p u t e t h e w e i g h t e n u m e r a t o r o f t h e d u a l c o d e , i . e . t h e b o n d s p a c e B o fKp ( o f d im e n s io n p - I ) . N o w th i s is a t r i v ia l t a s k : An y n - s u b s e t o f X o f t h e v e r t e x s e tV o f K p d e t e r m i n e s a u n i q u e p a r t i t i o n ( X , V \ X ) o f V w h i c h c o r r e s p o n d s t o t h e c u tc o n s i s t i n g o f t h e n(q - n ) e d g e s j o in in g a v e r t e x i n X to a v e r t e x in V \ X . A s e a c h c u ti s c o u n t e d t w i c e i n th i s w a y , t h e h o m o g e n e o u s w e i g h t e n u m e r a t o r o f B j u s t i s

z p ( x , y ) = ~ . = oH e n c e t h e h o m o g e n e o u s w e ig h t e n u m e r a t o r o f C i s

1w , (x , y) = ~ zp (x + y , x - y) .S u b s t i t u t i n g a n d p u t t i n g y = I imm e d ia t e ly g iv e s t h e d e s i r e d r e s u lt . [ ]


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Vol . 65 , 1995 A p rob lem in g raph ica l enum era t ion 463T h e p r e c e d i n g p r o o f o f R e a d ' s t h e o r e m is n o t o n l y s i g n if i ca n t ly s im p l e r ( a n d m o r e

c o n c e p t u a l ) t h a n t h e u s u a l o n e , b u t w e m a y a l s o u s e t h e s a m e a p p r o a c h t o o b t a i n t h eg e n e r a t i n g f u n c t i o n f o r t h e n u m b e r o f l a b e le d b i p a r t i t e e u l e r i a n g r a p h s w i t h p a r t so f m a n d n v e r t ic e s , re s p e c t i v e ly , a r e s u l t w h i c h s e e m s t o b e n e w ( a n d w o u l d b e u n p l e a s a n tt o p r o v e w i t h t he s t a n d a r d m e t h o d s o f g r a p h i c a l e n u m e r a t io n ) .

T h e o r e m 3 . T h e p o l y n o m i a l w " , , ( x ) w h i c h h a s a s t h e c o e f f i c i e n t o f x ~ t h e n u m b e ro f l a b e l e d b i p a r t i t e E u l e r i a n g r a p h s o n m + n v e r t i c e s ( w i t h t h e t w o p a r t s b e i n g o fs i z e s m a n d n , r e s p e c t i v e l y ) w i t h q e d g e s , is g i v e n b y

w " , , ( x ) 2 , , + , _ 1 (1 + k= o h :OP r o o f . N o t e t h a t t h e E u l e r i a n b i p a r t e g r a p h s o n m + n v e r t ic e s a r e p r e c i s e l y t h e e v e n

s u b g r a p h s o f t h e c o m p l e t e b i p a r ti t e g r a p h K i n , . . T h u s w , ,, , ( x) i s n o t h i n g b u t t h e ( n o n -h o m o g e n e o u s ) w e i g h t e n u m e r a t o r o f t h e e v e n g r a p h i c a l c o d e C = C ( K " , , ) . U s i n g t h eM a c W i l li a m s i de n ti ti es , w e m a y i n s te a d c o m p u t e t h e w e i g h t e n u m e r a t o r o f t h e d u a lc o d e , i .e . th e b o n d s p a c e B o f K " , , ( o f d i m e n s i o n m + n - 1 ). A g a i n , t h i s is a t ri v i a lt a s k : L e t t h e t w o s e t s i n t h e p a r t i t i o n o f t h e v e r t e x s e t V b e U a n d U ' , re s p e c t iv e l y .A n y p a i r ( X , Y ) c o n s i s t in g o f a k - s u b s e t X o f U a n d a n h - s u b se t Y o f U ' d e t e r m i n e sa u n i q u e p a r t i t i o n ( X w Y, ( U \ X ) u ( U ' \ Y ) ) o f V w h i c h c o r r e s p o n d s t o t h e c u t c o n s is t -i n g o f t h e k ( n - h ) + h ( m - k ) e d g e s e i t h e r j o i n i n g a v e r t e x i n X t o a v e r t e x i n U ' \ Y o ra v er te x in Y t o a v e rt ex i n U \ X . H e n c e t h e h o m o g e n e o u s w e i g h t e n u m e r a t o r o f Bj u s t i s

Zm, n( X ' y ) _ ~ x m n - k ( n - h ) - h ( m - k ) y k ( n - h ) + h ( m - k ) .k=O h= O

A c c o r d i n g t o t he M a c W i l li a m s i de n ti ti es , t h e h o m o g e n e o u s w e i g h t e n u m e r a t o r o f C i s1W,n, n ( X , y ) 2 m +n - 1 Z m , n ( X + y , X - - y ) .

S u b s t i t u t i n g a n d p u t t i n g y = i i m m e d i a t e l y g i v e s t h e d e s i r e d r e su l t. ORe fe r enc e s

[1] J. A. BONDYand U . S . R . MURTY , Gra ph theo ry wi th app lica tions . No r th Hol land , Am s te rdam1976.[2] J. G. BREDESONand S. L . HAKrMI, De cod ing of grap h theo ret ic codes . I EE E Trans . In form . Th.13, 348-349 (1967) .[3] S. L. HAKIM~and J. G. BREDESON,Gra ph theore t ic e r ro r -co r rec ting codes. IE E E T rans . In fo rm .Yh. 14, 584-591 (1968) .[4] E HARARYand E. PALIVa~R,Graph ica l enum era t ion . New York 1973 .[5 ] D. JUNGNICKELand S. A. VANSTONE,Graphical codes revisited. Submitted.[6] E J. MACWmLIAMSand N. J. A. SLOANE,T he theory o f e r ro r -co r rec t ing codes. N or th Hol land ,A m s t e r d a m 1 9 7 7.


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46 4 D. JUNGNICKEL a nd S. A . VANSTONE ARCH. MATH.[7 ] R . C . R E A D , E u l e r g r ap h s o n l ab e l ed n o d es . C a n ad . J. M a t h . 1 4, 4 8 2 -4 8 6 ( i 9 6 2 ) .[8 ] J . H . VA N L IN T , In t ro d u c t i o n t o co d i n g t h eo ry . B e r l i n -H e i d e l b e rg -N e w Y o rk 1 98 2 .

E i n g e g a n g e n a m 6 . 3 . 1 99 5A n s c h r i f t e n d e r A u t o r e n :D i e t e r J u n g n i c k e lL e h r s t u h l f i i r A n g e w a n d t e M a t h e m a t i k I IU n i v e r si t~ i t A u g s b u rgD - 8 6 1 3 5 A u g s b u r g

Sco t t A . V an s t o n eD e p a r t m e n t o f C o m b i n a t o r i c s a n d O p t i m i z a ti o nU n i v e r s i t y o f W a t e r lo oW a t e r lo o , O n t a r i o N 2 L 3 G IC a n a d a

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An application of coding theory to a problem